A194680 Number of k in [1,n] for which + > 1, where < > = fractional part, and r=3-sqrt(2).

1, 2, 3, 1, 1, 5, 2, 8, 4, 6, 5, 11, 2, 12, 8, 5, 3, 12, 8, 5, 13, 5, 23, 18, 16, 6, 6, 18, 21, 27, 7, 24, 12, 4, 23, 7, 26, 35, 31, 20, 41, 29, 9, 5, 19, 9, 47, 37, 5, 24, 24, 37, 50, 46, 22, 17, 9, 43, 53, 7, 55, 16, 44, 42, 4, 46, 13, 48, 46, 50, 63, 17, 57, 43, 38, 9

Offset: 1

Author

Clark Kimberling, Sep 01 2011

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A194679.

Programs

Mathematica

r = 3 - Sqrt[2]; z = 15;
p[x_] := FractionalPart[x]; f[x_] := Floor[x];
w[n_, k_] := p[r^n] + p[r^k] - p[r^n + r^k]
Flatten[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
  (* A194679 *)
TableForm[Table[w[n, k], {n, 1, z}, {k, 1, n}]]
s[n_] := Sum[w[n, k], {k, 1, n}]  (* A194680 *)
Table[s[n], {n, 1, 100}]
h[n_, k_] := f[p[n*r] + p[k*r]]
Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
  (* A194681 *)
TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
t[n_] := Sum[h[n, k], {k, 1, n}]
Table[t[n], {n, 1, 100}]   (* A194682 *)

PARI

for(n=1, 30, print1(round(sum(k=1, n, frac((3-sqrt(2))^n) + frac((3-sqrt(2))^k) - frac((3-sqrt(2))^n + (3-sqrt(2))^k))), ", ")) \\ G. C. Greubel, Feb 08 2018